On families of 2N-dimensional hypercomplex algebras suitable for digital signal processing

A survey of hypercomplex algebras suitable for DSP is presented. Generally applicable properties are obtained, including a paraunitarity condition for hypercomplex lossless systems. Algebras of dimension n = 2^N, N ∈ ℤ, are classified by generation methods, constituting families. Two algebra families, which hold commutative and associative properties for arbitrary N, are examined in more detail: The 2^N-dimensional hyperbolic numbers and tessarines. Since these non-division algebras possess zero divisors, orthogonal decomposition of hypercomplex numbers is investigated in general.